The paper studies the $K$-theoretic invariants of the crossed product
$C^{*}$-algebras associated with an important family of homeomorphisms
of the tori $\mathbb{T}^{n}$ called Furstenberg transformations.
Using the Pimsner-Voiculescu theorem, we prove that given $n$, the
$K$-groups of those crossed products, whose corresponding $n\times n$
integer matrices are unipotent of maximal degree, always have the same
rank $a_{n}$. We show using the theory developed here that a claim
made in the literature about the torsion subgroups of these $K$-groups
is false. Using the representation theory of the simple Lie algebra
$\frak{sl}(2,\mathbb{C})$, we show that, remarkably, $a_{n}$ has a
combinatorial significance. For example, every $a_{2n+1}$ is just the
number of ways that $0$ can be represented as a sum of integers
between $-n$ and $n$ (with no repetitions). By adapting an argument
of van Lint (in which he answered a question of ErdÅs), a simple,
explicit formula for the asymptotic behavior of the sequence
$\{a_{n}\}$ is given. Finally, we describe the order structure of the
$K_{0}$-groups of an important class of Furstenberg crossed products,
obtaining their complete Elliott invariant using classification
results of H. Lin and N. C. Phillips.

Let $\mathfrak{A}$ be a $C^{*}$-algebra with real rank zero which has
the stable weak cancellation property. Let $\mathfrak{I}$ be an ideal
of $\mathfrak{A}$ such that $\mathfrak{I}$ is stable and satisfies the
corona factorization property. We prove that
$
0 \to \mathfrak{I} \to \mathfrak{A} \to \mathfrak{A} / \mathfrak{I} \to 0
$
is a full extension if and only if the extension is stenotic and
$K$-lexicographic. {As an immediate application, we extend the
classification result for graph $C^*$-algebras obtained by Tomforde
and the first named author to the general non-unital case. In
combination with recent results by Katsura, Tomforde, West and the
first author, our result may also be used to give a purely
$K$-theoretical description of when an essential extension of two
simple and stable graph $C^*$-algebras is again a graph
$C^*$-algebra.}

We study projections in the corona algebra of $C(X)\otimes K$, where K
is the $C^*$-algebra of compact operators on a separable infinite
dimensional Hilbert space and $X=[0,1],[0,\infty),(-\infty,\infty)$,
or $[0,1]/\{ 0,1 \}$. Using BDF's essential codimension, we determine
conditions for a projection in the corona algebra to be liftable to a
projection in the multiplier algebra. We also determine the
conditions for two projections to be equal in $K_0$, Murray-von
Neumann equivalent, unitarily equivalent, or homotopic. In light of
these characterizations, we construct examples showing that the
equivalence notions above are all distinct.

Let $A$ be a separable amenable purely infinite simple \CA which
satisfies the Universal Coefficient Theorem. We prove that $A$ is
weakly semiprojective if and only if $K_i(A)$ is a countable
direct sum of finitely generated groups ($i=0,1$). Therefore, if
$A$ is such a \CA, for any $\ep>0$ and any finite subset ${\mathcal
F}\subset A$ there exist $\dt>0$ and a finite subset ${\mathcal
G}\subset A$ satisfying the following: for any contractive
positive linear map $L: A\to B$ (for any \CA $B$) with $
\|L(ab)-L(a)L(b)\|
Keywords:weakly semiprojective, purely infinite simple $C^*$-algebrasCategories:46L05, 46L80

We show that the Elliott invariant is a classifying invariant for the
class of $C^*$-algebras that are simple unital infinite dimensional
inductive limits of finite direct sums of building blocks of the form
$$
\{f \in C(\T) \otimes M_n : f(x_i) \in M_{d_i}, i = 1,2,\dots,N\},
$$
where $x_1,x_2,\dots,x_N \in \T$, $d_1,d_2,\dots,d_N$ are integers
dividing $n$, and $M_{d_i}$ is embedded unitally into $M_n$.
Furthermore we prove existence and uniqueness theorems for
$*$-homomorphisms between such algebras and we identify the range of
the invariant.

Let $\Gamma$ be a torsion free lattice in $G=\PGL(3, \mathbb{F})$ where
$\mathbb{F}$ is a nonarchimedean local field. Then $\Gamma$ acts freely
on the affine Bruhat-Tits building $\mathcal{B}$ of $G$ and there is an
induced action on the boundary $\Omega$ of $\mathcal{B}$. The crossed
product $C^*$-algebra $\mathcal{A}(\Gamma)=C(\Omega) \rtimes \Gamma$
depends only on $\Gamma$ and is classified by its $K$-theory. This article
shows how to compute the $K$-theory of $\mathcal{A}(\Gamma)$ and of the
larger class of rank two Cuntz-Krieger algebras.

We give a description of the monoid of Murray-von Neumann equivalence
classes of projections for multiplier algebras of a wide class of
$\sigma$-unital simple $C^\ast$-algebras $A$ with real rank zero and stable
rank one. The lattice of ideals of this monoid, which is known to be
crucial for understanding the ideal structure of the multiplier
algebra $\mul$, is therefore analyzed. In important cases it is shown
that, if $A$ has finite scale then the quotient of $\mul$ modulo any
closed ideal $I$ that properly contains $A$ has stable rank one. The
intricacy of the ideal structure of $\mul$ is reflected in the fact
that $\mul$ can have uncountably many different quotients, each one
having uncountably many closed ideals forming a chain with respect to
inclusion.

For a dense $G_\delta$ set of real parameters $\theta$ in $[0,1]$
(containing the rationals) it is shown that the group $K_0 (A_\theta
\rtimes_\sigma \mathbb{Z}_4)$ is isomorphic to $\mathbb{Z}^9$, where
$A_\theta$ is the rotation C*-algebra generated by unitaries $U$, $V$
satisfying $VU = e^{2\pi i\theta} UV$ and $\sigma$ is the Fourier
automorphism of $A_\theta$ defined by $\sigma(U) = V$, $\sigma(V) =
U^{-1}$. More precisely, an explicit basis for $K_0$ consisting of
nine canonical modules is given. (A slight generalization of this
result is also obtained for certain separable continuous fields of
unital C*-algebras over $[0,1]$.) The Connes Chern character $\ch
\colon K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4) \to H^{\ev} (A_\theta
\rtimes_\sigma \mathbb{Z}_4)^*$ is shown to be injective for a dense
$G_\delta$ set of parameters $\theta$. The main computational tool in
this paper is a group homomorphism $\vtr \colon K_0 (A_\theta
\rtimes_\sigma \mathbb{Z}_4) \to \mathbb{R}^8 \times \mathbb{Z}$
obtained from the Connes Chern character by restricting the
functionals in its codomain to a certain nine-dimensional subspace of
$H^{\ev} (A_\theta \rtimes_\sigma \mathbb{Z}_4)$. The range of $\vtr$
is fully determined for each $\theta$. (We conjecture that this
subspace is all of $H^{\ev}$.)

Giordano, Putnam and Skau showed that the transformation group
$C^*$-algebra arising from a Cantor minimal system is an $AT$-algebra,
and classified it by its $K$-theory. For approximately inner
automorphisms that preserve $C(X)$, we will determine their classes in
the Ext and OrderExt groups, and introduce a new invariant for the
closure of the topological full group. We will also prove that every
automorphism in the kernel of the homomorphism into the Ext group is
homotopic to an inner automorphism, which extends Kishimoto's result.

A simple $\C^*$-algebra is constructed for which the Murray-von
Neumann equivalence classes of projections, with the usual
addition---induced by addition of orthogonal projections---form the
additive semi-group
$$
\{0,2,3,\dots\}.
$$
(This is a particularly simple instance of the phenomenon of
perforation of the ordered $\K_0$-group, which has long been known in
the commutative case---for instance, in the case of the
four-sphere---and was recently observed by the second author in the
case of a simple $\C^*$-algebra.)

Let $A_\theta$ denote the rotation algebra---the universal $C^\ast$-algebra
generated by unitaries $U,V$ satisfying $VU=e^{2\pi i\theta}UV$, where
$\theta$ is a fixed real number. Let $\sigma$ denote the Fourier
automorphism of $A_\theta$ defined by $U\mapsto V$, $V\mapsto U^{-1}$,
and let $B_\theta = A_\theta \rtimes_\sigma \mathbb{Z}/4\mathbb{Z}$ denote
the associated $C^\ast$-crossed product. It is shown that there is a
canonical inclusion $\mathbb{Z}^9 \hookrightarrow K_0(B_\theta)$ for each
$\theta$ given by nine canonical modules. The unbounded trace functionals
of $B_\theta$ (yielding the Chern characters here) are calculated to obtain
the cyclic cohomology group of order zero $\HC^0(B_\theta)$ when
$\theta$ is irrational. The Chern characters of the nine modules---and more
importantly, the Fourier module---are computed and shown to involve techniques
from the theory of Jacobi's theta functions. Also derived are explicit
equations connecting unbounded traces across strong Morita equivalence, which
turn out to be non-commutative extensions of certain theta function equations.
These results provide the basis for showing that for a dense $G_\delta$ set
of values of $\theta$ one has $K_0(B_\theta)\cong\mathbb{Z}^9$ and is
generated by the nine classes constructed here.

Let $A$ be a simple $C^*$-algebra
with real rank zero, stable rank one and weakly
unperforated $K_0(A)$ of countable rank. We show that
a monomorphism $\phi\colon C(S^2) \to A$ can be approximated
pointwise by homomorphisms from $C(S^2)$ into $A$ with
finite dimensional range if and only if certain index
vanishes. In particular, we show that every homomorphism
$\phi$ from $C(S^2)$ into a UHF-algebra can be approximated
pointwise by homomorphisms from $C(S^2)$ into the UHF-algebra
with finite dimensional range. As an application, we show
that if $A$ is a simple $C^*$-algebra of real rank zero
and is an inductive limit of matrices over $C(S^2)$ then
$A$ is an AF-algebra. Similar results for tori are also
obtained. Classification of ${\bf Hom}\bigl(C(X),A\bigr)$
for lower dimensional spaces is also studied.